The Philosophical magazine; a journal of theoretical, experimental and applied physics . % = 0, (3) and 2«cos2iS« +2/3 sin2(^,S;S-(«--/3^) sin 2(^x8(^,-0. (4)But J(/0 ^ S ^ «-^sin(^,cos^\ ,r-. a»=/j4 a; /• • • ^^^ Substituting these values in (3) and (4) and eliminating S0ibetween them we get («2E-^-^F + «Aicot(/)i)S«+«-(F-E + Aitan,)Sy3=0,) \ (^)or x\S« + BS/3 = 0. J Now the mean curvature of a surface of revolution of mini-mum area has been proved by Lindelof to be the same as. 38 Mr. A. W. Riicker on the Critical Mean Curvaturethat of a circle of radius (« + /9). Hence A-B J^ [cc+^y- A {oc+
The Philosophical magazine; a journal of theoretical, experimental and applied physics . % = 0, (3) and 2«cos2iS« +2/3 sin2(^,S;S-(«--/3^) sin 2(^x8(^,-0. (4)But J(/0 ^ S ^ «-^sin(^,cos^\ ,r-. a»=/j4 a; /• • • ^^^ Substituting these values in (3) and (4) and eliminating S0ibetween them we get («2E-^-^F + «Aicot(/)i)S«+«-(F-E + Aitan,)Sy3=0,) \ (^)or x\S« + BS/3 = 0. J Now the mean curvature of a surface of revolution of mini-mum area has been proved by Lindelof to be the same as. 38 Mr. A. W. Riicker on the Critical Mean Curvaturethat of a circle of radius (« + /9). Hence A-B J^ [cc+^y- A {oc+^y Hence the mean curvature has in general a critical valuewhen A-B = 0. First confining ourselves to the case in which the prin-cipal ordinate is a maximum, and ^^ and 6 are less than7r/2, it is evident that, since F is always >E, B is alwayspositive Also, by (5), 1^K die whence, since dF/clK is positive and sin <^i cos ^i is positive,«^E —/3-F is positive, and therefore A is positive also. Further,B can only vanish if 0i = O ; and none of the terms in A or Bbecome infinite unless i = 0 or 7r/2, cases which it will beseen hereafter it is unnecessary to , (A-B)/a2 = 2E-F(l+cos^^) + 2AiCot2^i = 0 . (7) is a relation which must be satisfied by ^j and d when themean curvature has a critical value for changes in the formof the surface which take place, subject to the conditions thatthe radii of the rincrs and the distance between them areconsta
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